**Linear Counting (Group 3)**

**Introduction**

The names of the numbers are introduced in Group 1 with the ‘Formation of Numbers’ when the Director checks the correspondence of the quantities with the card, she uses the conventional names for the tens and combinations of tens and units. As the child works with the decimal system she may shown an interest in the names and the Group 3 activities can be introduced, otherwise we give the conventional names to the child before she leaves the Casa at five to six years.

There are three groups of names

- Names for a combination of a ten and units one to nine, these are ‘teens’
- Names for a group of ten – ten, twenty, thirty etc.
- Names for figures from the tens category and a unit, these help with linear counting

**11 -19 Teens (Beads only)**

**Material Description:**

9 bars of ten Golden Beads, one of each coloured bead bar, representing the units 1 – 9 and box for each of these. For older children give forty five single beads, or use these only in the second Period with younger children

Each quantity is distinguished by a different colour

- red
- green
- pink
- yellow
- light blue
- grey or violet
- white
- violet or brown
- dark blue

A small felt cloth on a Working Mat

**Display**

With the material for linear counting

**Presentation:**

Introduction

- Show the material to the child; removing one bead bar at a time ask the child to identify the number of beads in each bar at random, make reference to the colour and provide a three period lesson if necessary
- Sort the bead bars into an isosceles triangle, known as a Bead Stair

Three Period Lesson

First Period

- Take the bar of ten and place the unit to the right of it, adjacent to the first bead
- Count the beads
- Say, “One ten and one are also called ‘eleven’”
- Repeat the sequence for ‘twelve’ and ‘thirteen’

Second Period

- Mix all the previously introduced bead bars and invite the child to make the numbers, continue mixing to maintain the child’s interest

Third Period

- Make a quantity and ask the child to name it
- Begin each subsequent three Period Lesson counting up from eleven

**Criteria of Perfection (Control of Error):**

- The child’s own sound knowledge of the numbers 1 to 10 and their numerical order acts as a guide

**Direct Aim:**

- The coloured bead bars show clearly the separate entities from 1 to 9, in combination with the tens they show the child that numbers 11 to 19 are made of ten AND a number 1 to 9
- To learn the names of the quantities 11-19
- To learn the sequence of the numbers 11-19

**Age at Presentation:**

Four and a half years onwards, when the child knows the numbers 1 to 10 well, after the ‘Formation of Numbers’, parallel to, or after the ‘Decimal System’ and before the child leaves the Casa.

**Footnote:**

- Explain that, ‘teen’ indicated ‘ten’, so ‘sixteen’ means ‘six and ten’
- When the child has learnt the numbers 11 to 19 choose a moment to verify by forming a quantity and asking the child to name it. Ask the child to make her on quantities and, also make the quantities 11 to 19 in order, ask her to count forward and backwards
- The ‘bar of ten’ is placed vertically so that the units can be built parallel with it, almost forming a new ten. Placing the ‘bar of ten’ horizontally can be done to prepare for linear counting
- The child can also use loose Golden Beads for her activity

**11 -19 Teens Board (cards only)**

**Material Description:**

- Two wooden slated boards with five partitions each, on nine of the partitions a large 10 is written in black, the last partition is empty
- Loose wooden cards with the digits 1 to 9 which slide into the boards from the right, covering the ‘0’
- Working Mat

**Display**

The box is kept after the beads 11-19

**Presentation:**

- Place the boards on the Working Mat, and the cards at random nearby
- Give the symbols in a three Period Lesson

First Period

- Slip the ‘1’ over the 0’ of the first ten, saying, ”This is eleven”, do the same with ‘twelve’ and ‘thirteen’

Second Period

- Ask the child to identify previously introduced numbers by moving the cards and mixing them, ask the child to make a number using the cards and boards

Third Period

- Make a number with the cards and ask the child to identify it

;

- Continue till 19 on the same day or later, depending on the child
- When complete ask the child to count forwards and backwards

**Criteria of Perfection (Control of Error):**

- The child’s own knowledge of the numbers from 1 to 10 and their numerical order acts as a guide.

**Direct Aim:**

- To introduce the child to the symbols for the numbers 11 to 19 and to continue to associate their names.

**Age at Presentation:**

Four and a half years of age, after presenting the quantities

**Footnote:**

- Explain that ‘teen’ indicated ‘ten’, so ‘sixteen’ means ‘six and ten’
- When the child has learnt the numbers 11 to 19 choose a moment to verify by forming a quantity and asking the child to name it. Ask the child to make her on quantities and, also make the quantities 11 to 19 in order, ask her to count forward and backwards.

**11 -19 Teens Boards and Beads**

**Material Description:**

- Short Bead Stair
- 9 bars of ten in a box
- Teen boards
- Working Mat
- For the exercises have one ten card in blue and 1 to 9 unit cards in green.

**Presentation:**

- Lay out the boards on the mat, with the cards placed at random to the right, and the beads, in a Bead Stair, to the left, the tens in their box
- Place a ‘bar of ten’ and a bead to form eleven to the left of the top section of the board and slip the card of ‘1’ over the ‘0’ to form the figure ’11’
- Place a ‘bar of ten’ and two beads to form twelve to the left of the top section of the board and slip the card of ‘2’ over the ‘0’ to form the figure ’12’
- Let the child continue till she reaches 19
- When she completes ask her to count forwards and backwards

;

;

**Exercises:**

- Let the child make the numbers with the beads and cards in order and randomly
- Use the large cards instead of the boards
- Also use the loose golden beads instead of the bars

**Criteria of Perfection (Control of Error):**

- The child’s own knowledge of the numbers from 1 to 10 and their numerical order acts as a guide

**Direct Aim:**

- Continued association of the quantity, name and symbol for 11 to 19; to reinforce the sequence 11 to 19

**Age at Presentation:**

Four and a half years of age, after presenting the quantities and their symbols.

**11 – 99 Tens Boards and Beads**

**Material Description:**

- Two wooden slated boards with five partitions each, on nine of the partitions are the tens numbers, 10, 20, 30 etc, the last partition is empty, later use Large cards
- Loose wooden cards with the digits 1 to 9 which slide into the boards from the right, covering the ‘0’
- 45 bars of ten in a box
- Working Mat

**Presentation:**

Names of the Tens

- Layout the boards on the working mat, place the boxes with the beads to the left. Place one bar of ten by the first ten
- Indicate ’20’ beneath and the child names however she likes, say, “Twenty also means two tens”, continue this till you reach ’90’
- Give a Three Period Lesson for any of the names the child is unfamiliar with.

**Criteria of Perfection (Control of Error):**

- The child’s own knowledge of the numbers and their order will guide her

**Direct Aim:**

- To learn the conventional names of the tens from 10 to 90 and to realise that 20 is the same as two tens etc
- To realise how the numbers progress from one ten to the next and to see the pattern in making and counting numbers up to 99

**Age at Presentation:**

Four and a half onwards, after the presentation of teens

**Second Activity for Tens Boards and Beads **

On a separate day give the following presentation

**Material Description:**

- Two wooden slated boards with five partitions each, on nine of the partitions are the tens numbers, 10, 20, 30 etc, the last partition is empty, later use Large cards
- Loose wooden cards with the digits 1 to 9 which slide into the boards from the right, covering the ‘0’
- 45 bars of ten in a box and box with nine Golden Bead units
- Working Mat

**Presentation**

This activity helps the child to count from 11 to 99

- Keep the sets of cards in a stack and the beads together to the left of the boards
- To write ’11’ place a ‘bar of ten’ and a unit to the left of the boards and slot in the card of ‘1’ over the ‘0’, then add another bead to make ’12’ and replace the ‘1’ card with the ‘2’ card.
- Continue to 19, say, “If we had one more bead we would get a ten (indicate the loose beads) so we would have two tens, two tens are also called ‘twenty’”.
- Put two ‘bars of ten’ together by the second partition and change the cards of the one above
- Continue to 99, composing the words verbally, in beads and symbols

**Exercises:**

- The child’s own exercises with the material
- The child forms her own numbers using the bead material and the Large Cards

**Criteria of Perfection (Control of Error):**

- The child’s own knowledge of the numbers and their order will guide her

**Direct Aim:**

- To learn the conventional names of the tens from 10 to 90 and to realise that 20 is the same as two tens etc
- To realise how the numbers progress from one ten to the next and to see the pattern in making and counting numbers up to 99

**Age at Presentation:**

Four and a half onwards, after the presentation of teens

**Writing the Numbers 1 to 100**

**Material Description:**

Samples of written numbers, coloured and graphite pencils

Sheets of squared paper

**Presentation:**

- Show a few samples to the child
- Let the child copy the numbers from a simple sample and later write independently

**Exercises:**

- After the child has written the numbers say numbers and let her find them and cross them diagonally to form a pattern in a contrast colour
- Two children can do the above activity together
- Give number squares with ‘Grid References’, the numbers 1-9 horizontally and 10 to 90 vertically, the child can find where ‘five’ meets ‘twenty’ to discover the number ‘twenty five’

**Criteria of Perfection (Control of Error):**

- The child’s own knowledge of the numbers and their order will guide her

**Direct Aim:**

- To learn the conventional names of the tens from 10 to 90 and to realise that 20 is the same as two tens etc
- To realise how the numbers progress from one ten to the next and to see the pattern in making and counting numbers up to 99

**Age at Presentation:**

After the tens boards and beads

**Footnote:**

- When the child tires of writing the numbers give her a completed number square so she can play games with more speed

**Linear Counting: the 100 and 1,000 chains**

The teen and tens boards introduce counting up to ninety-nine.

The Linear Activity reinforces this and extends the child’s ability to count beyond 100, which the child is drive to do, doing so builds her self esteem. The sensorial difference between a ‘square of a 100’ and a ‘cube of a thousand’, first given with the ‘Decimal System’ is reinforced. Linear counting which is mechanical and restful is securely explored here.

**100 bead chain**

**Material Description:**

100 chain

- The 100 chain made up of ten ‘bars of ten’ joined together, one square of 100 in Golden Beads
- Small arrows 1-9 in green, 10 to 90 in blue and 100 in red, in a box
- A tray, a felt mat and a Working Mat

**Presentation:**

- Lay out the 100 chain mat
- Go to the bead material with the child and show her how to hold the hundreds chain
- Lay the 100s chain on the mat and bring the other materials
- Fold the 100s chain into a square and compare it with the ‘square of a hundred’, first placing it next to the folded chain and then superimposing it, then place it to one side.
- Pull the chain slowly into a straight line, showing that ‘rods of ten’ can be put together to form a hundreds chain
- Read the arrows out loud with the child
- Place the arrows of each category in separate groups
- Count and place the green arrows beside the first nine beads
- Place the blue arrow ’10’ by the tenth bead continue with the blue arrows counting each bead
- Finally place the ‘square of a hundred’ below the last bead
- Read all the arrows forwards and backwards

**Exercises:**

- The child’s own activity with the material
- Mix the arrows, give the child one at a time and let her read and place it by the chain. She counts and places the arrows
- Without using the chain, ask the child to give you the arrow and then the next number until you build a list of arrows in sequence

**1,000 bead chain**

**Material Description:**

- The 1,000 chain made up of a hundred ‘bars of ten’, a larger ring connects them at each 100, ten ‘bars of ten’ joined together
- Ten square of 100 in Golden Beads and one cube of one thousand
- Small arrows 1-9 in green, 10 to 90 in blue, 100 to 900 in red, and 1,000 in green in a box
- A tray, a felt mat and Working Mats

**Presentation:**

- Lay out the 1000 chain mat
- Go to the bead material with the child and show her how to hold the thousands chain, by unhooking the larger rings and placing one at a time on the index of your left finger
- Lay the 1000s chain on the mat, laying it out fully and bring the other materials
- Fold the 1000s chain into ten squares of one hundred and compare them with the ‘square of a hundred’, first superimposing them, then stacking the ‘squares of a hundred’ to allow the child to compare it to the ‘cube of a thousand’, lastly place it to one side.
- Pull the chain slowly into a straight line, showing that ‘rods of ten’ can be put together to form a thousands chain
- Introduce the arrows read some out loud with the child and group them, non-sequentially into their different categories
- Count and place the green arrows beside the first nine beads
- Place the blue arrow ’10’ by the tenth bead continue with the blue arrows counting each bead till 90.
- Put a red arrow by 100 and continue till 900
- Place the green arrow of 1,000 by the final bead
- Finally place the ‘square of a hundred’ after each hundred and the ‘cube of a thousand’ at the end of the chain
- Read all the arrows forwards and backwards

**Exercises:**

- The child’s own activity with the material
- Mix the arrows, give the child one at a time and let her read and place it by the chain. She counts and places the arrows
- Without using the chain, ask the child to give you the arrow and then the next number until you build a list of arrows in sequence

**Criteria of Perfection (Control of Error):**

- The labels must fit the last of each bead bar as the child counts the beads. Thus the child easily sees at the end of a chain if she has made an error in counting

**Direct Aim:**

- To give the child practice in linear counting
- To reinforce the sequence of numbers up to 1,000
- To consolidate the child’s knowledge of counting
- To show the striking difference between the chain of 100 and 1,000, which is the difference between 10
^{2}and 10^{3}

**Indirect Aim:**

- Preparation for multiplication
- Preparation for squaring and cubing

**Age at Presentation:**

Five to five and a half years

**Skip Counting**

**Material Description:**

Cabinet of cubes, squares and chains (suspended 20 cm above the floor, on the wall, with a shelf below for boxes of labels)

Three red unit beads:1, 2 squared and 1 cubed

Long and short chains of two in green; green labels marked 1,2,4 for short chain; 1,2,4,6, and 8 for long chain.

Long and short chains of three in pink; pink labels marked 1,2,3,6 and 9 for short chain; to 27 for long chain.

Long and short chains of four in yellow; yellow labels marked 1,2,3,4, 8 and each four till 16 for short chain; to 64 for long chain.

Long and short chains of five in light blue; light blue labels marked 1,2, 3, 4, 5, 10 and each five till 25 for short chain; to 125 for long chain.

Long and short chains of six in mauve; mauve labels marked 1, 2, 3, 4, 5, 6, 12 and each six till 36 for short chain; to 216 for long chain.

Long and short chains of seven in white; white labels marked 1, 2, 3, 4, 5, 6, 7,14 and each seven till 27 for short chain; to 343 for long chain.

Long and short chains of eight in brown; brown labels marked 1, 2, 3, 4, 5, 6, 7, 8, 16 and each eight till 64 for short chain; to 512 for long chain.

Long and short chains of nine in blue labels marked 1, 2, 3, 4, 5, 6, 7, 8, 9, 18 and each nine till 81 for short chain; to 729 for long chain.

**Presentation:**

Preliminary Activity

- Take the mat, short chains and prepare the Geometric Cabinet
- Let the child make shapes with each short chain, the chain of 3 forms a triangle etc and match them to the Geometrical Shapes of the Cabinet
- Compare all the chains by laying them in parallel lines, shortest to longest

The Short Chain

- Choose one short chain and lay out the mat
- Fetch the chain and then the corresponding square and arrows
- Fold the chain into squares and compare by superimposing the square and putting it to one side
- Introduce the arrows, placing them in any order
- Slowly unfold the chain into a straight line, count the chain in a linear fashion, touching each bead and placing the corresponding arrow adjacent
- Place the square at the end of the chain, by the last bead
- Invite the child to skip count forwards and backwards

The Long Chain

- Choose one long chain and lay out the mat
- Fetch the chain and then the corresponding square and arrows
- Fold the chain into squares and compare by superimposing the square and putting it to one side
- Introduce the arrows, placing them in any order
- Slowly unfold the chain into a straight line, count the chain in a linear fashion, touching each bead and placing the corresponding arrow adjacent
- Place a square each time that many have been counted, then take all the squares build a cube and compare it with the cube that corresponds to that chain, to show equivalence.
- Invite the child to skip count forwards and backwards

**Exercises:**

- The child’s own activities with the short and long chains
- Take any arrow and ask her to find it’s place on the chain
- Take any set of arrows and arrange from small to large without the chains
- The child does the chains of any power together

**Criteria of Perfection (Control of Error):**

- The labels must fit the last of each bead bar as the child counts the beads. Thus the child easily sees at the end of a chain if she has made an error in counting.

**Direct Aim:**

- To give the child further experience in linear counting
- When reading the labels, the child is given the opportunity to ‘skip’ count and thereby indirectly preparing for the multiplication of numbers

**Indirect Aim:**

- Indirect preparation for multiples

**Age at Presentation:**

Five to five and a half years, parallel to linear counting

**Footnote:**

Indirectly prepares for the further realisation of the difference in quantity between the square and the cube of each number. The child has not been given the idea of powers of numbers, yet the interest the child has in counting at this stage enables the Director to demonstrate (first sensorially) by stretching out the beads in two parallel lines, and then by counting, the number of beads in each chain, the difference in quantity between the square and the cube.

The chains also bring into evidence, taken in sensorially through the sight, the growth from the linear aspect of the unit becoming a ‘rods of ten’ and then into a square, and later into a cube. The child also witnesses the relationship between at the squares of that chain and the ‘cube of a thousand’. And between one square and all the squares, one cube and all the cubes.

If the child asks what is meant by the 6^{2} or the 6^{3} on the back of the final arrow for the short and long chain respectively take either the square or the cube and have her count the numbers in each dimension, revealing that we have ‘six in two ways’, which make a “six squared”, so 6^{2} or ‘six three ways’, which make a cube so we call it “six cubed” 6^{3}.